Group homology is the one-object case
For a small category A and a functor T from it into an abelian category, the homology of A with coefficients in T is the left derived functor of the colimit — equivalently, of the left Kan extension to the terminal category. When A has one object and all morphisms invertible it is a group, and the construction returns group homology exactly. The nerve makes the whole picture topological: this homology is the homology of a classifying space.
Learning objectives
- Define the homology of a small category with functor coefficients.
- Recover group homology as the one-object case.
- Describe the nerve and the classifying space.
- State the spectral sequences relating a functor between categories.
Section 01The definition
where K is the functor to the terminal category. The functor category Fun(A, C) is abelian when C is, with enough projectives when A is small, so the derived functors exist.
| Category A | Homology obtained |
|---|---|
| A group, viewed as one object with invertible morphisms | Group homology |
| A monoid | Monoid homology |
| A poset | Homology of the order complex |
| A discrete category | Direct sum of the coefficients; nothing higher |
| The simplex category | Simplicial homology of the coefficient object |
| A groupoid | Direct sum over connected components of group homologies |
A functor from a one-object groupoid is exactly a module over the group, and the colimit is exactly the coinvariants. Left deriving the coinvariants is the definition of group homology, so the two constructions coincide by definition rather than by coincidence.
Section 02The nerve and classifying space
The nerve of A is the simplicial set whose n-simplices are chains of n composable morphisms. Its geometric realisation is the classifying space BA, and
For a group this is the usual classifying space BG, an Eilenberg–MacLane space K(G, 1), which is why group homology and the homology of BG agree.
The simplicial structure of the nerve, applied to a group, is precisely the bar resolution. The apparently ad hoc formula of the group-cohomology stream is the simplicial structure of a classifying space, written algebraically.
Section 03Functors between categories
A functor F: A → B induces restriction on coefficients and a comparison of homologies, and the Grothendieck construction gives a spectral sequence
Quillen's Theorem A
If every comma category F ↓ b is contractible, F induces a homotopy equivalence of classifying spaces — and hence an isomorphism on homology.
Cofinality
A cofinal functor induces an isomorphism on colimits and on their derived functors, which is how many computations are reduced to a smaller indexing category.
The LHS sequence again
For a surjection of groups viewed as a functor of one-object categories, the spectral sequence above is the Lyndon–Hochschild–Serre sequence.
ReferenceFrequently asked questions
Why must the category be small?
Because the functor category must have enough projectives and the colimits must exist, both of which require a set of objects. For large categories the constructions may fail or require universe conventions.
Is this the same as simplicial homology?
It is computed by the simplicial object given by the nerve, so yes in that sense. The categorical formulation makes the coefficient functor available, which the purely simplicial picture does not naturally provide.
What does contractibility of a comma category mean here?
That its classifying space is contractible, so it contributes nothing above degree 0 to the spectral sequence. Quillen's Theorem A turns that vanishing into a statement that the two categories have the same homology — the standard reduction technique in algebraic K-theory.
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